Block Method for Solving the Laplace Equation and for Constructing Conformal Mappings

This book presents a new, efficient numerical-analytical method for solving the Laplace equation on an arbitrary polygon. This method, called the approximate block method, overcomes indicated difficulties and has qualitatively more rapid convergence than well-known difference and variational-difference methods. The block method also solves the complicated problem of approximate conformal mapping of multiply-connected polygons onto canonical domains with no preliminary information required. The high-precision results of calculations carried out on the computer are presented in an abundance of tables substantiating the exponential convergence of the block method and its strong stability concerning the rounding-off of errors.

Approximate Block Method for Solving the Laplace Equation on PolygonsSetting up a Mixed Boundary Value Problem for the Laplace Equation on a PolygonA Finite Covering of a Polygon by Blocks of Three TypesRepresentation of the Solution of a Boundary Value Problem on BlocksAn Algebraic ProblemThe Main Result - Theorem on the Convergence of the Block MethodProofs of Theorem and LemmasThe Stability and the Labor Content of Computations Required by the Block MethodApproximation of a Conjugate Harmonic Function on BlocksNeumann's ProblemThe Case of Arbitrary Analytic Mixed Boundary ConditionsApproximate Block Method of Conformal Mapping of Polygons onto Canonical DomainsApproximate Conformal Mapping of a Simply-Connected Polygon onto a DiskBasic Harmonic FunctionsApproximate Conformal Mapping of a Multiply-Connected Polygon onto a Plane with Cuts along Parallel Line SegmentsApproximate Conformal Mapping of a Multiply-Connected Polygon onto a Ring with Cuts along the Arcs of Concentric CirclesDevelopment and Application of the Approximate Block Method for Conformal Mapping of Simply-Connected and Doubly-Connected DomainsApproximate Conformal Mapping of Some Polygons onto a StripScheme of Constructing a Conformal Mapping of a Doubly-connected Domain onto a RingMapping a Square Frame onto a RingMapping a Square with a Circular Hole Using Circular Lune BlockRepresentation of a Harmonic Function on a RingUsing a Block-Ring for Mapping Domain (18.1) onto a RingA Block-BridgeLimit CasesMapping a Disk with an Elliptic Hole or with a Retro-Section onto a RingMapping a Disk with a Regular Polygonal HoleMapping the Exterior of a Parabola with a Hole onto a RingApproximate Conformal Mapping of Domains with a Periodic Structure by the Block MethodMapping a Domain of the Type of Half-Plane with a Periodic Structure onto a Half-planeMapping a Domain of the Type of Strip with a Periodic Structure onto a StripMapping the Exterior of a Lattice of Ellipses onto the Exterior of a Lattice of PlatesReferencesIndex

Evgenii A. Volkov is a professor at the Steklov Mathematical Institute in Moscow, Russia.